What are partial orders?

Some articles on partial orders, order, partial order:

Order Dimension Two
... The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different ... That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x, y of distinct elements is comparable in ... If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions ...
Birkhoff's Representation Theorem - Functoriality
... as stated above, is a correspondence between individual partial orders and distributive lattices ... However, it can also be extended to a correspondence between order-preserving functions of partial orders and bounded homomorphisms of the ... Let 2 denote the partial order on the two-element set {0, 1}, with the order relation 0 < 1, and (following Stanley) let J(P) denote the distributive ...
Coxeter Group - Partial Orders
... Using reduced words one may define three partial orders on the Coxeter group, the (right) weak order, the absolute order and the Bruhat order (named ... An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some ... In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment ...
Linear Extension - Related Results
... The algorithmic problem of constructing a linear extension of a partial order on a finite set, represented by a directed acyclic graph with the set's elements as its vertices, is known as topological sorting ... extension, the problem of counting all linear extensions of a finite partial order is #P-complete however, it may be estimated by a fully polynomial-time randomized approximation scheme ... Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are semiorders ...

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